Molecular Orbitals
An approach to bonding in which orbitals
encompass the entire molecule, rather than
being localized between atoms.
Molecular Orbitals
Molecular orbitals result from the
combination of atomic orbitals.
Since orbitals are wave functions, they can
combine either constructively (forming a
bonding molecular orbital), or destructively
(forming an antibonding molecular orbital).
Molecular Orbitals
Molecular orbitals form when atomic
orbitals with similar energies and proper
symmetry can overlap.
Atomic orbitals with differing energies or the
wrong spatial orientation (orthogonal) do not
combine, and are called non-bonding orbitals.
Need for MO Theory
Valence bond theory fails to explain the
bonding in many simple molecules.
The oxygen molecule has a bond length and
strength consistent with a double bond, and it
contains two unpaired electrons.
Need for MO Theory
Valence bond theory predicts the double
bond, but not the paramagnetism of oxygen.
: :
: :
O=O
Need for MO Theory
Resonance is another example of the
limitations of valence bond theory. Bond
lengths and strengths are intermediate between
single, double or triple bonds.
Molecular orbital theory is often a better
approach to use with molecules that have
extended π systems.
Molecular Orbital Theory
In order to simplify things, we’ll consider the
interaction of the orbitals containing valence
electrons to create molecular orbitals.
The wave functions of hydrogen atom A and
hydrogen atom B can interact either
constructively or destructively.
Molecular Orbital Theory
Constructively:
Ψ(σ) or Ψ+ = (1/√2 ) [φ(1sa) + φ(1sb) ]
Destructively:
Ψ(σ*) or Ψ- = (1/√2 ) [φ(1sa) - φ(1sb) ]
Molecular Orbital Theory
The bonding orbital
results in increased
electron density between
the two nuclei, and is of
lower energy than the
two separate atomic
orbitals.
Molecular Orbital Theory
The antibonding
orbital results in a node
between the two nuclei,
and is of greater energy
than the two separate
atomic orbitals.
Molecular Orbital Theory
The result is an
energy level diagram with
the bonding orbital
occupied by a pair of
electrons. The filling of
the lower molecular
orbital indicates that the
molecule is stable
compared to the two
individual atoms.
Molecular Orbital Theory
+
-
+
The bonding orbital is
sometimes given the
notation σg, where the g
stands for gerade, or
symmetric with respect
to a center of inversion.
The signs on the molecular orbitals indicate the sign of
the wave function, not ionic charge.
Molecular Orbital Theory
+
-
+
The anti-bonding orbital
is sometimes given the
notation σu, where the u
stands for ungerade, or
asymmetric with respect
to a center of inversion.
The signs on the molecular orbitals indicate the sign of
the wave function, not ionic charge.
Rules for Combining Atomic
Orbitals
1.
2.
The number of molecular orbitals = the
number of atomic orbitals combined.
The strength of the bond depends upon the
degree of orbital overlap.
Experimental Evidence
Photoelectron spectroscopy (PES) is a
technique in which a beam of ultraviolet light
with an energy of 21 eV is used to irradiate
molecules.
The energy is high enough to eject electrons.
The kinetic energy of the emitted electrons is
measured, and used to determine the energy
level of the electron.
Experimental Evidence
The technique allows
for the measurement
of specific ionization
energies (I). Each
ionization energy
represents the
removal of an
electron from a
specific molecular
orbital.
Experimental Evidence
Electrons in
lower energy levels
require more energy
to be removed, and
are ejected with less
kinetic energy.
hνo = I + Ekinetic
Period 2 Diatomic Molecules
For the second period, assume that, due to a
better energy match, s orbitals combine with s
orbitals, and p orbitals combine with p orbitals.
The symmetry of p orbitals permits end-onend overlap along the bond axis, or side-by-side
overlap around, but not along, the internuclear
axis.
MOs using p orbitals
+ -
-
-
+
+
-
With the x axis as the bond axis, the px
orbitals may combine constructively or
destructively. The result is a σ bonding orbital
and a σ anti-bonding orbital.
MOs using p orbitals
+ -
+
-
+ -
-
The designation σ indicates symmetric
electron density around the internuclear (x) axis.
The + and – signs indicate the sign of the wave
function, and not electrical charges.
MOs using p orbitals
+ -
+
-
+ -
-
Some texts will use the symmetry
designations of g (gerade) or u (ungerade) instead
of indicating bonding or anti-bonding.
MOs using p orbitals
+ -
+
-
+ -
-
σg
For these orbitals, the bonding orbital is
gerade, or symmetric around the bond axis.
MOs using p orbitals
+ -
+
-
+ -
-
σu
σg
For these orbitals, the anti-bonding orbital is
asymmetric about the bond axis, and is
designated as σu. Note that the designations of u
or g do not correlate with bonding or antibonding.
π Molecular Orbitals
+
-
+
+
side-by-side
overlap
-
The orbital overlap side-by-side is less than
that of overlap along the bond axis (end-onend). As a result, the bonding orbital will be
higher in energy than the previous example.
π Molecular Orbitals
+
-
+
+
side-by-side
overlap
-
π orbitals are asymmetric with respect to the
bond axis. There is electron density surrounding
the bond axis, with a node along the internuclear
axis.
π Molecular Orbitals
+
-
+
+
side-by-side
overlap
-
πu
Some texts use the subscripts g and u instead
of bonding and anti-bonding. In this example,
the bonding orbital is ungerade, or asymmetric
about a center of symmetry.
π Molecular Orbitals
+
-
+
+
side-by-side
overlap
-
πg
πu
The anti-bonding orbital is gerade, or
symmetric about a center of symmetry.
Molecular Orbital Diagram
This is a molecular
orbital energy level
diagram for the p
orbitals. Note that the
σ bonding orbital is
lowest in energy due to
the greater overlap
end-on-end.
σu
πg
2p
πu
σg
2p
Molecular Orbital Diagram
The alternate
notation is provided
on the right side of the
energy level diagram.
σu
πg
2p
πu
σg
2p
Molecular Orbital Diagrams
1.
2.
3.
4.
Electrons preferentially occupy molecular
orbitals that are lower in energy.
Molecular orbitals may be empty, or contain
one or two electrons.
If two electrons occupy the same molecular
orbital, they must be spin paired.
When occupying degenerate molecular
orbitals, electrons occupy separate orbitals
with parallel spins before pairing.
Molecular Orbital Diagrams
Although molecular orbitals form from inner
(core) electrons as well as valence electrons,
many molecular orbital diagrams include only
the valence level.
Molecular Orbital Diagrams
For O2, there
will be a total of
12 valence
electrons that
must be placed in
the diagram.
Molecular Orbital Diagrams
For O2, there
will be a total of
12 valence
electrons that
must be placed in
the diagram.
Molecular Orbital Diagrams
2p
2s
2p
2s
For O2, there
will be a total of
12 valence
electrons that
must be placed in
the diagram.
MO Diagram for O2
σ*u
π*g
2p
2s
πu
σg
σ*u
σg
2p
2s
The molecular
orbital diagram for
oxygen shows two
unpaired electrons,
consistent with
experimental data.
Bond Order
Bond order is an indicator of the bond
strength and length. A bond order of 1 is
equivalent to a single bond. Fractional bond
orders are possible.
The bond order of the molecule =
(# e- in bonding orbtls) - (# e- in anti-bonding orbtls)
2
2
MO Diagram for O2
The bond order of
O2 is:
σ*u
π*g
2p
2s
πu
σg
σ*u
σg
8-4 = 2
2
2p
2s
This is consistent
with a double
bond.
MO Diagram for O2
σ*u
π*g
2p
2s
πu
σg
σ*u
σg
2p
2s
This energy level
diagram works well
for atoms in which
the 2s and 2p levels
are fairly far apart.
These are the
elements at the right
of the table: O, F and
Ne.
Experimental Evidence
Oxygen is paramagnetic, consistent with having
two unpaired electrons. In addition, photoelectron
spectroscopy (PES) can be used for determining orbital
energies in molecules. The molecule is bombarded
with UV or X-rays to remove an electron from the
molecule. The kinetic energy of the emitted electron is
measured and subtracted from the incident radiation to
determine the binding energy of the electron.
Photoelectron Spectroscopy
The result is a spectrum of absorptions which are
correlated to the molecular orbitals of the molecule. In
addition, electrons ejected from bonding orbitals show
more vibrational energy levels than electrons emitted
from anti-bonding or non-bonding orbitals.
MO diagram for Li through N
The elements on the left side of period 2
have a fairly small energy gap between the 2s
and 2p orbitals. As a result, interaction between
s and p orbitals is possible. This can be viewed
in different ways.
MO diagram for Li through N
In some approaches, the s orbital on one atom
interacts with the p orbital on another. The interaction
can be constructive or destructive.
MO diagram for Li through N
In another approach, the s and p orbitals on
the same atom interact in what is called orbital
mixing.
Either approach yields the same result. The
σ bonding and anti-bonding orbitals are raised in
energy due to the interaction with a p orbital.
MO diagram for Li through N
σ*u
π*g
σg
πu
σ*u
σg
MO diagram for N2
σ*u
π*g
σg
πu
σ*u
σg
N2 has 10
valence
electrons.
Experimental Evidence
σg
πu
σ*u
The photoelectronic
spectrum of nitrogen is
consistent with a
molecular orbital
approach.
Electrons emitted
from bonding orbitals
show vibrational
excitations.
Experimental Evidence
σ*u
π*g
σg
πu
σg
πu
σ*u
σ*u
σg
Heteronuclear Diatomic
Molecules
The more electronegative atom will have
orbitals of lower energy, and therefore
contribute more to the bonding orbitals.
The less electronegative atom has orbitals of
higher energy, and contributes more to the antibonding orbitals.
Rules for Combining Atomic
Orbitals
For heteronuclear molecules:
1. The bonding orbital(s) will reside
predominantly on the atom of lower orbital
energy (the more electronegative atom).
2. The anti-bonding orbital(s) will reside
predominantly on the atom with greater orbital
energy (the less electronegative atom).
HF
The 2s and 2px orbitals
on fluorine interact with
the 1s orbital on hydrogen.
The py and pz orbitals
on fluorine lack proper
symmetry to interact with
hydrogen, and remain as
non-bonding orbitals.
HF
The anti-bonding
orbital resides primarily on
the less electronegative
atom (H).
Note that the
subscripts g and u are not
used, as the molecule no
longer has a center of
symmetry.
Carbon monoxide
In carbon
monoxide, the bonding
orbitals reside more on
the oxygen atom, and
the anti-bonding
orbitals reside more on
the carbon atom.
Carbon monoxide
CO is a highly
reactive molecule with
transition metals.
Reactivity typically
arises from the highest
occupied molecular
orbital (HOMO), when
donating electrons.
Carbon monoxide
When acting as an
electron pair acceptor,
the lowest
unoccupied molecular
orbital (LUMO), is
significant.
Carbon monoxide
When acting as an
electron pair donor,
the highest occupied
molecular orbital
(HOMO), is
significant.
The highest
occupied molecular
orbital of CO is a
molecular orbital
which puts
significant electron
density on the
carbon atom.
The lowest
unoccupied
molecular orbital of
CO is the π* orbitals.
The lobes of the
LUMO are larger on
the carbon atom
than on the oxygen
atom.
CO as a Ligand
Carbon monoxide is known as a σ donor and
a π acceptor ligand. It donates electrons from
its HOMO to form a sigma bond with the
metal.
CO as a Ligand
Carbon monoxide accepts electrons from
filled d orbitals on the metal into its antibonding
(LUMO) orbital.
CO as a Ligand
This phenomenon is called back bonding. The
increased electron density in the antibonding orbitals of
CO causes an increase in the C-O bond length and a
decrease in its stretching frequency.
MOs for Larger Molecules
Group theory is usually used to develop
molecular orbital diagrams and drawings of
more complicated molecules. When a central
atom is bonded to several atoms of the same
element (H2O, BF3, or PtCl42-], group theory can
be used to analyze the symmetry of the orbitals
of the non-central atoms, and then combine
them with the appropriate orbitals of the central
atom.
MOs for Larger Molecules
The orbitals of the non-central atoms are
called group orbitals. In considering a simple
example, H2O, we obtain group orbitals using
the two 1s orbitals on the hydrogen atoms.
The characters for the
group orbitals is obtained by
considering each hydrogen
as a spherical 1s orbital.
They remain in position for
identity, are exchanged
during rotation, remain in
place for σxz (the molecular
plane), and are exchanged
for σyz.
Group Orbitals of Water
Γred and its irreducible representations are:
Group Orbitals of Water
The A1 representation has both 1s orbitals with
positive wave functions: Ha+Hb.
The B1 representations is Ha+Hb.
Group Orbitals of Water
These group orbitals are combined with orbitals
on oxygen that have the same symmetry.
Group Orbitals of Water
The 2s and 2pz orbital on oxygen have
A1 symmetry, the 2px orbital has B1 symmetry,
and the 2py has B2 symmetry.
Molecular Orbitals of Water
Since the 2py orbital on oxygen doesn’t
match the symmetry of the group orbitals of
hydrogen, it will remain non-bonding. The
other orbitals on oxygen will combine with the
appropriate group orbitals to form bonding and
antibonding molecular orbitals.
MOs for Larger Molecules
Group theory is usually used to develop
molecular orbital diagrams and drawings of
more complicated molecules. A simplified
example will be shown for the π bonding of
benzene.
π Bonding of Benzene
Benzene belongs to point group D6h. In
determining the orbital combinations for π
bonding, we need to obtain Гπ by looking only
at the pz orbitals on each carbon atom.
We need only consider
those orbitals on carbon
atoms that remain in place
for a given symmetry
operation.
π Bonding of Benzene
C″2
C′2
{
z axis
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ
π Bonding of Benzene
C′2
C″2
{
z axis
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ 6
0
0
0
-2
0
0 0
0
-6
0
2
π Bonding of Benzene
C′2
C″2
{
z axis
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ 6
0
0
0
-2
0
0 0
0
This reduces to: B2g + E1g + A2u + E2u
-6
0
2
π Bonding of Benzene
Гπ: B2g + E1g + A2u + E2u
Group theory can be used to draw each of
the π molecular orbitals. Molecular orbitals with
fewer nodes are lower in energy (more bonding),
and those with more nodes are higher in energy
(more antibonding).
π Bonding of Benzene
Гπ: B2g + E1g + A2u + E2u
A2u fully bonding and
lowest in energy
E1g degenerate
bonding orbitals
with one node
π Bonding of Benzene
Гπ: B2g + E1g + A2u + E2u
E2u degenerate
largely antibonding orbitals
with two nodes
B2g fully antibonding orbital
with three nodes
π Bonding of Benzene
B2g
E2u
E1g
A2u
Molecular Orbitals of Complexes
Group theory is also used to construct
molecular orbital diagrams for the complexes of
metal atoms or ions. The symmetry
combinations of the atomic orbitals on the
ligands are determined, and then “matched”
with appropriate atomic orbitals on the central
metal. Both σ and π bonding between the metal
and ligands can be considered.